




"""simple docstring"""




import math



def        a__					(							snake_case__							)		->   bool:

			if 1 < number < 4:
						# 2 and 3 are primes
						return True
			elif number < 2 or number % 2 == 0 or number % 3 == 0:
						# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
						return False

			# All primes number are in format of 6k +/- 1
			for i in range(5			,					int(math.sqrt(snake_case__							) + 1							)			,					6							):
						if number % i == 0 or number % (i + 2) == 0:
									return False
			return True



def        a__					(							snake_case__ = 0.1							)		->   int:
			lowerCamelCase      =   3
			lowerCamelCase      =   3

			while primes / (2 * j - 1) >= ratio:
						for i in range(j * j + j + 1			,					(j + 2) * (j + 2)			,					j + 1							):
									primes += is_prime(snake_case__							)
						j += 2
			return j


if __name__ == "__main__":
					import doctest

					doctest.testmod()









"""simple docstring"""




from __future__ import annotations

from decimal import Decimal

from numpy import array



def        a__					(							snake_case__							)		->   list[list[float]]:
			lowerCamelCase      =   Decimal

			# Check if the provided matrix has 2 rows and 2 columns
			# since this implementation only works for 2x2 matrices
			if len(snake_case__							) == 2 and len(matrix[0]							) == 2 and len(matrix[1]							) == 2:
						# Calculate the determinant of the matrix
						lowerCamelCase      =   float(
						    d(matrix[0][0]							) * d(matrix[1][1]							) - d(matrix[1][0]							) * d(matrix[0][1]							)							)
						if determinant == 0:
									raise ValueError("""This matrix has no inverse."""							)

						# Creates a copy of the matrix with swapped positions of the elements
						lowerCamelCase      =   [[0.0, 0.0], [0.0, 0.0]]
						lowerCamelCase ,  lowerCamelCase      =   matrix[1][1], matrix[0][0]
						lowerCamelCase ,  lowerCamelCase      =   -matrix[1][0], -matrix[0][1]

						# Calculate the inverse of the matrix
						return [
						    [(float(d(snake_case__							)							) / determinant) or 0.0 for n in row] for row in swapped_matrix
						]
			elif (
			    len(snake_case__							) == 3
			    and len(matrix[0]							) == 3
			    and len(matrix[1]							) == 3
			    and len(matrix[2]							) == 3
			):
						# Calculate the determinant of the matrix using Sarrus rule
						lowerCamelCase      =   float(
						    (
						        (d(matrix[0][0]							) * d(matrix[1][1]							) * d(matrix[2][2]							))
						        + (d(matrix[0][1]							) * d(matrix[1][2]							) * d(matrix[2][0]							))
						        + (d(matrix[0][2]							) * d(matrix[1][0]							) * d(matrix[2][1]							))
						    )
						    - (
						        (d(matrix[0][2]							) * d(matrix[1][1]							) * d(matrix[2][0]							))
						        + (d(matrix[0][1]							) * d(matrix[1][0]							) * d(matrix[2][2]							))
						        + (d(matrix[0][0]							) * d(matrix[1][2]							) * d(matrix[2][1]							))
						    )							)
						if determinant == 0:
									raise ValueError("""This matrix has no inverse."""							)

						# Creating cofactor matrix
						lowerCamelCase      =   [
						    [d(0.0							), d(0.0							), d(0.0							)],
						    [d(0.0							), d(0.0							), d(0.0							)],
						    [d(0.0							), d(0.0							), d(0.0							)],
						]
						lowerCamelCase      =   (d(matrix[1][1]							) * d(matrix[2][2]							)) - (
						    d(matrix[1][2]							) * d(matrix[2][1]							)
						)
						lowerCamelCase      =   -(
						    (d(matrix[1][0]							) * d(matrix[2][2]							)) - (d(matrix[1][2]							) * d(matrix[2][0]							))
						)
						lowerCamelCase      =   (d(matrix[1][0]							) * d(matrix[2][1]							)) - (
						    d(matrix[1][1]							) * d(matrix[2][0]							)
						)
						lowerCamelCase      =   -(
						    (d(matrix[0][1]							) * d(matrix[2][2]							)) - (d(matrix[0][2]							) * d(matrix[2][1]							))
						)
						lowerCamelCase      =   (d(matrix[0][0]							) * d(matrix[2][2]							)) - (
						    d(matrix[0][2]							) * d(matrix[2][0]							)
						)
						lowerCamelCase      =   -(
						    (d(matrix[0][0]							) * d(matrix[2][1]							)) - (d(matrix[0][1]							) * d(matrix[2][0]							))
						)
						lowerCamelCase      =   (d(matrix[0][1]							) * d(matrix[1][2]							)) - (
						    d(matrix[0][2]							) * d(matrix[1][1]							)
						)
						lowerCamelCase      =   -(
						    (d(matrix[0][0]							) * d(matrix[1][2]							)) - (d(matrix[0][2]							) * d(matrix[1][0]							))
						)
						lowerCamelCase      =   (d(matrix[0][0]							) * d(matrix[1][1]							)) - (
						    d(matrix[0][1]							) * d(matrix[1][0]							)
						)

						# Transpose the cofactor matrix (Adjoint matrix)
						lowerCamelCase      =   array(snake_case__							)
						for i in range(3							):
									for j in range(3							):
												lowerCamelCase      =   cofactor_matrix[j][i]

        # Inverse of the matrix using the formula (1/determinant) * adjoint matrix
						lowerCamelCase      =   array(snake_case__							)
						for i in range(3							):
									for j in range(3							):
												inverse_matrix[i][j] /= d(snake_case__							)

        # Calculate the inverse of the matrix
						return [[float(d(snake_case__							)							) or 0.0 for n in row] for row in inverse_matrix]
			raise ValueError("""Please provide a matrix of size 2x2 or 3x3."""							)



